Problem: Simplify and expand the following expression: $ \dfrac{5}{a - 6}- \dfrac{4}{4a + 8}+ \dfrac{3}{a^2 - 4a - 12} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $4$ out of denominator in the second term: $ \dfrac{4}{4a + 8} = \dfrac{4}{4(a + 2)}$ We can factor the quadratic in the third term: $ \dfrac{3}{a^2 - 4a - 12} = \dfrac{3}{(a - 6)(a + 2)}$ Now we have: $ \dfrac{5}{a - 6}- \dfrac{4}{4(a + 2)}+ \dfrac{3}{(a - 6)(a + 2)} $ The least common multiple of the denominators is: $ (a - 6)(a + 2)$ In order to get the first term over $(a - 6)(a + 2)$ , multiply by $\dfrac{4(a + 2)}{4(a + 2)}$ $ \dfrac{5}{a - 6} \times \dfrac{4(a + 2)}{4(a + 2)} = \dfrac{20(a + 2)}{(a - 6)(a + 2)} $ In order to get the second term over $(a - 6)(a + 2)$ , multiply by $\dfrac{a - 6}{a - 6}$ $ \dfrac{4}{4(a + 2)} \times \dfrac{a - 6}{a - 6} = \dfrac{4(a - 6)}{(a - 6)(a + 2)} $ In order to get the third term over $(a - 6)(a + 2)$ , multiply by $\dfrac{4}{4}$ $ \dfrac{3}{(a - 6)(a + 2)} \times \dfrac{4}{4} = \dfrac{12}{(a - 6)(a + 2)} $ Now we have: $ \dfrac{20(a + 2)}{(a - 6)(a + 2)} - \dfrac{4(a - 6)}{(a - 6)(a + 2)} + \dfrac{12}{(a - 6)(a + 2)} $ $ = \dfrac{ 20(a + 2) - 4(a - 6) + 12} {(a - 6)(a + 2)} $ Expand: $ = \dfrac{20a + 40 - 4a + 24 + 12}{4a^2 - 16a - 48} $ $ = \dfrac{16a + 76}{4a^2 - 16a - 48}$ Simplify: $ = \dfrac{4a + 19}{a^2 - 4a - 12}$